Average Error: 12.2 → 0.3
Time: 2.9m
Precision: 64
Internal Precision: 384
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;v \cdot v + v \le 4.148278326517786 \cdot 10^{+101}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{0.125}{1 - v \cdot v} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \cdot \left(3 - \left(v + v\right)\right)\right) \cdot \left(1 + v\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + \left(3 - 4.5\right)\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25\right) + \frac{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left(\frac{0.125}{v} + 0.125\right)}{v}\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 2 regimes

  2. if (+ (* v v) v) < 4.148278326517786e+101

    1. Initial program 8.5

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
    2. Using strategy rm

    3. Applied flip--8.5

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{\color{blue}{\frac{1 \cdot 1 - v \cdot v}{1 + v}}}\right) - 4.5\]

    4. Applied associate-/r/8.5

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 \cdot 1 - v \cdot v} \cdot \left(1 + v\right)}\right) - 4.5\]

    5. Applied simplify0.7

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{0.125}{1 - v \cdot v} \cdot \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(3 - \left(v + v\right)\right)\right)\right)} \cdot \left(1 + v\right)\right) - 4.5\]
    6. Using strategy rm

    7. Applied associate-*r*0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{0.125}{1 - v \cdot v} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \cdot \left(3 - \left(v + v\right)\right)\right)} \cdot \left(1 + v\right)\right) - 4.5\]

    if 4.148278326517786e+101 < (+ (* v v) v)

    1. Initial program 17.4

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]

    2. Taylor expanded around inf 16.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right) - \left(0.125 \cdot \frac{{w}^{2} \cdot {r}^{2}}{v} + 0.125 \cdot \frac{{w}^{2} \cdot {r}^{2}}{{v}^{2}}\right)\right)}\right) - 4.5\]

    3. Applied simplify0.3

      \[\leadsto \color{blue}{\left(\left(\frac{2}{r \cdot r} + \left(3 - 4.5\right)\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25\right) + \frac{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left(\frac{0.125}{v} + 0.125\right)}{v}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))